## Meyer sets and their duals.(English)Zbl 0880.43008

Moody, Robert V. (ed.), The mathematics of long-range aperiodic order. Proceedings of the NATO Advanced Study Institute, Waterloo, Ontario, Canada, August 21–September 1, 1995. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 489, 403-441 (1997).
A Delone set $$S$$ (i.e. a point set in $$\mathbb{R}^n$$ which is both uniformly discrete and relatively dense) is a Meyer set iff also $$S-S$$ is Delone. This characterization is due to J. Lagarias and probably the simplest presently known. These sets were introduced by Y. Meyer in the context of harmonic analysis. Later they re-appeared in the study of aperiodic, but ordered, structures such as quasicrystals. This paper reviews the various characterizations and the key properties of Meyer sets, with emphasis on their rôle as generalizations of lattices and their connection to mathematical quasicrystals.
For the entire collection see [Zbl 0861.00015].

### MSC:

 43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 11K60 Diophantine approximation in probabilistic number theory 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)

### Keywords:

tilings; Delone set; Meyer set; harmonic analysis; quasicrystals; lattices